Partial Differential Equations/Poisson's equation

Provided that , we will through distribution theory prove a solution formula, and for domains with boundaries satisfying a certain property we will even show a solution formula for the boundary value problem. We will also study solutions of the homogenous Poisson's equation

The solutions to the homogenous Poisson's equation are called harmonic functions.

In section 2, we had seen Leibniz' integral rule, and in section 4, Fubini's theorem. In this section, we repeat the other theorems from multi-dimensional integration which we need in order to carry on with applying the theory of distributions to partial differential equations. Proofs will not be given, since understanding the proofs of these theorems is not very important for the understanding of this wikibook. The only exception will be theorem 6.3, which follows from theorem 6.2. The proof of this theorem is an exercise.

Theorem 6.1: (Dominated convergence theorem)

Let be a sequence of functions such that

for a , which is independent of . Then

Theorem 6.2: (Divergence theorem)

Let a compact set with smooth boundary. If is a vector field, then

, where is the outward normal vector.

Theorem 6.3: (Multi-dimensional integration by parts)

Let a compact set with smooth boundary. If is a function and is a vector field, then

If the Gamma function is shifted by 1, it is an interpolation of the factorial (see exercise 2):

As you can see, in the above plot the Gamma function also has values on negative numbers. This is because what is plotted above is some sort of a natural continuation of the Gamma function which one can construct using complex analysis.

Definition and theorem 6.7:

The -dimensional spherical coordinates, given by

are a diffeomorphism. The determinant of the Jacobian matrix of , , is given by

Proof:

Theorem 6.8:

The volume of the -dimensional ball with radius , is given by

Proof:

Theorem 6.9:

The area of the surface of the -dimensional ball with radius (i. e. the area of ) is given by

The surface area and the volume of the -dimensional ball with radius are related to each other "in a differential way" (see exercise 3).

We show that is locally integrable. Let be compact. We have to show that

is a real number, which by lemma 6.11 is equivalent to

is a real number. As compact in is equivalent to bounded and closed, we may choose an such that . Without loss of generality we choose , since if it turns out that the chosen is , any will do as well. Then we have

For ,

For ,

, where we applied integration by substitution using spherical coordinates from the first to the second line.

2.

We calculate some derivatives of (see exercise 5):

For , we have

For , we have

For all , we have

3.

We show that

Let and be arbitrary. In this last step of the proof, we will only manipulate the term . Since , has compact support. Let's define

Since the support of

, where is the characteristic function of .

The last integral is taken over (which is bounded and as the intersection of the closed sets and closed and thus compact as well). In this area, due to the above second part of this proof, is continuously differentiable. Therefore, we are allowed to integrate by parts. Thus, noting that is the outward normal vector in of , we obtain

Let's furthermore choose . Then

.

From Gauß' theorem, we obtain

, where the minus in the right hand side occurs because we need the inward normal vector. From this follows immediately that

We can now calculate the following, using the Cauchy-Schwartz inequality:

Proof: We choose as an orientation the border orientation of the sphere. We know that for , an outward normal vector field is given by . As a parametrisation of , we only choose the identity function, obtaining that the basis for the tangent space there is the standard basis, which in turn means that the volume form of is

Now, we use the normal vector field to obtain the volume form of :

We insert the formula for and then use Laplace's determinant formula:

As a parametrisation of we choose spherical coordinates with constant radius .

We calculate the Jacobian matrix for the spherical coordinates:

We observe that in the first column, we have only the spherical coordinates divided by . If we fix , the first column disappears. Let's call the resulting matrix and our parametrisation, namely spherical coordinates with constant , . Then we have:

Recalling that

, the claim follows using the definition of the surface integral.

Theorem 6.13:

Let be a function. Then

Proof:

We have , where are the spherical coordinates. Therefore, by integration by substitution, Fubini's theorem and the above formula for integration over the unit sphere,

From first coordinate transformation with the diffeomorphism and then applying our formula for integration on the unit sphere twice, we obtain:

From first differentiation under the integral sign and then Gauss' theorem, we know that

Case 1: If is harmonic, then we have

, which is why is constant. Now we can use the dominated convergence theorem for the following calculation:

Therefore for all .

With the relationship

, which is true because of our formula for , we obtain that

, which proves the first formula.

Furthermore, we can prove the second formula by first transformation of variables, then integrating by onion skins, then using the first formula of this theorem and then integration by onion skins again:

This shows that if is harmonic, then the two formulas for calculating , hold.

Case 2: Suppose that is not harmonic. Then there exists an such that . Without loss of generality, we assume that ; the proof for will be completely analoguous exept that the direction of the inequalities will interchange. Then, since as above, due to the dominated convergence theorem, we have

Since is continuous (by the dominated convergence theorem), this is why grows at , which is a contradiction to the first formula.

The contradiction to the second formula can be obtained by observing that is continuous and therefore there exists a

This means that since

and therefore

, that

and therefore, by the same calculation as above,

This shows (by proof with contradiction) that if one of the two formulas hold, then is harmonic.

Definition 6.16:

A domain is an open and connected subset of .

For the proof of the next theorem, we need two theorems from other subjects, the first from integration theory and the second from topology.

Theorem 6.17:

Let and let be a function. If

then for almost every .

Theorem 6.18:

In a connected topological space, the only simultaneously open and closed sets are the whole space and the empty set.

We will omit the proofs.

Theorem 6.19:

Let be a domain and let be harmonic. If there exists an such that

, then is constant.

Proof:

We choose

Since is open by assumption and , for every exists an such that

By theorem 6.15, we obtain in this case:

Further,

, which is why

Since

, we have even

By theorem 6.17 we conclude that

almost everywhere in , and since

is continuous, even

really everywhere in (see exercise 6). Therefore , and since was arbitrary, is open.

Also,

and is continuous. Thus, as a one-point set is closed, lemma 3.13 says is closed in . Thus is simultaneously open and closed. By theorem 6.18, we obtain that either or . And since by assumtion is not empty, we have .

Theorem 6.18:

Let be a domain and let be harmonic. If there exists an such that

, then is constant.

Proof: See exercise 7.

Corollary 6.20:

Let be a bounded domain and let be harmonic on and continuous on . Then

Proof:

Theorem 6.20:

Let be open and be a harmonic function, let and let such that . Then

Proof:

What we will do next is showing that every harmonic function is in fact automatically contained in . But before we do so, we need another multiindex definition; using the definition of the usual binomial coefficient, we define a binomial coefficient for multiindices.

Definition 6.21:

If are two -dimensional multiindices, we define the binomial coefficient of over as:

With these multiindex definitions, we are able to write down a more general version of Leibniz' product rule for derivatives. But in order to prove this rule, we need another lemma first.

Lemma 6.22:

If and , where the is at the -th place, we have

for arbitrary multiindices .

Proof:

For the ordinary binomial coefficients for natural numbers, we had the formula

.

Therefore,

We also define less or equal relation on the set of multi-indices.

Definition 6.23:

Let be two -dimensional multiindices. We define to be less or equal than iff

For , there are vectors such that neither nor . For , the following two vectors are examples for this:

This example can be generalised to higher dimensions (see exercise 8).

This is the general product rule:

Theorem 6.24:

Let and let . Then

Proof:

We prove the claim by induction over .

1.

We start with the induction base . Then the formula just reads

, and this is true. Therefore, we have completed the induction base.

2.

Next, we do the induction step. Let's assume the claim is true for all such that . Let now such that . Let's choose such that (we may do this because ). We define again , where the is at the -th place. Then we have, due to the the Schwarz theorem and the not-generalized Leibniz product rule:

Now we may use the linearity of derivation and the induction hypothesis to obtain:

Then, here comes a key ingredient for the proof: Noticing that

and

, we notice that we are allowed to shift indices in the first of the two above sums, and furthermore simplify both sums with the rule

.

Therefore, we obtain:

Now we just sort the sum differently, and then apply our observation

,

which we made immediately after we defined the binomial coefficients, as well as the observations that

where in , and (these two rules may be checked from the definition of )

, to find in conclusion:

Theorem 6.25: Let be open, and let be harmonic. Then . Furthermore, for all , there is a constant depending only on the dimension and such that for all and such that

Proof:

Definition 6.26:

Let be a sequence of harmonic functions, and let be a function. converges locally uniformly to iff

Theorem 6.27:

Let be open and let be harmonic functions such that the sequence converges locally uniformly to a function . Then also is harmonic.

Proof:

Definition 6.28:

Theorem 6.29: (Arzelà-Ascoli) Let be a set of continuous functions, which are defined on a compact set . Then the following two statements are equivalent:

(the closure of ) is compact

is bounded and equicontinuous

Proof:

Definition 6.30:

Theorem 6.31:

Let be a locally uniformly bounded sequence of harmonic functions. Then it has a locally uniformly convergent subsequence.

Let be a domain. Let be the Green's kernel of Poisson's equation, which we have calculated above, i.e.

, where denotes the surface area of .

Suppose there is a function which satisfies

Then the Green's function of the first kind for for is defined as follows:

is automatically a Green's function for . This is verified exactly the same way as veryfying that is a Green's kernel. The only additional thing we need to know is that does not play any role in the limit processes because it is bounded.

A property of this function is that it satisfies

The second of these equations is clear from the definition, and the first follows recalling that we calculated above (where we calculated the Green's kernel), that for .

Here shall be continuous on . Then the following holds: The unique solution for this problem is given by:

Proof: Uniqueness we have already proven; we have shown that for all Dirichlet problems for on bounded domains (and the unit ball is of course bounded), the solutions are unique.

Therefore, it only remains to show that the above function is a solution to the problem. To do so, we note first that

Let be arbitrary. Since is continuous in , we have that on it is bounded. Therefore, by the fundamental estimate, we know that the integral is bounded, since the sphere, the set over which is integrated, is a bounded set, and therefore the whole integral must be always below a certain constant. But this means, that we are allowed to differentiate under the integral sign on , and since was arbitrary, we can directly conclude that on ,

Furthermore, we have to show that , i. e. that is continuous on the boundary.

To do this, we notice first that

This follows due to the fact that if , then solves the problem

and the application of the representation formula.

Furthermore, if and , we have due to the second triangle inequality:

In addition, another application of the second triangle inequality gives:

Let then be arbitrary, and let . Then, due to the continuity of , we are allowed to choose such that

.

In the end, with the help of all the previous estimations we have made, we may unleash the last chain of inequalities which shows that the representation formula is true:

Proof: For this proof, the very important thing to notice is that the formula for inside is nothing but the solution formula for the Dirichlet problem on the ball. Therefore, we immediately obtain that is superharmonic, and furthermore, the values on don't change, which is why . This was to show.